Theorem bj-clel3gALT 34781

Description: Alternate proof of clel3g 3575. (Contributed by BJ, 1-Sep-2024.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
bj-clel3gALT	⊢ (𝐵 ∈ 𝑉 → (𝐴 ∈ 𝐵 ↔ ∃𝑥(𝑥 = 𝐵 ∧ 𝐴 ∈ 𝑥)))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Allowed substitution hint:   𝑉(𝑥)

Proof of Theorem bj-clel3gALT

Step	Hyp	Ref	Expression
1		elisset 2833	. . . 4 ⊢ (𝐵 ∈ 𝑉 → ∃𝑥 𝑥 = 𝐵)
2	1	biantrurd 536	. . 3 ⊢ (𝐵 ∈ 𝑉 → (𝐴 ∈ 𝐵 ↔ (∃𝑥 𝑥 = 𝐵 ∧ 𝐴 ∈ 𝐵)))
3		19.41v 1950	. . 3 ⊢ (∃𝑥(𝑥 = 𝐵 ∧ 𝐴 ∈ 𝐵) ↔ (∃𝑥 𝑥 = 𝐵 ∧ 𝐴 ∈ 𝐵))
4	2, 3	bitr4di 292	. 2 ⊢ (𝐵 ∈ 𝑉 → (𝐴 ∈ 𝐵 ↔ ∃𝑥(𝑥 = 𝐵 ∧ 𝐴 ∈ 𝐵)))
5		eleq2 2840	. . . . 5 ⊢ (𝑥 = 𝐵 → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ 𝐵))
6	5	bicomd 226	. . . 4 ⊢ (𝑥 = 𝐵 → (𝐴 ∈ 𝐵 ↔ 𝐴 ∈ 𝑥))
7	6	pm5.32i 578	. . 3 ⊢ ((𝑥 = 𝐵 ∧ 𝐴 ∈ 𝐵) ↔ (𝑥 = 𝐵 ∧ 𝐴 ∈ 𝑥))
8	7	exbii 1849	. 2 ⊢ (∃𝑥(𝑥 = 𝐵 ∧ 𝐴 ∈ 𝐵) ↔ ∃𝑥(𝑥 = 𝐵 ∧ 𝐴 ∈ 𝑥))
9	4, 8	bitrdi 290	1 ⊢ (𝐵 ∈ 𝑉 → (𝐴 ∈ 𝐵 ↔ ∃𝑥(𝑥 = 𝐵 ∧ 𝐴 ∈ 𝑥)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538  ∃wex 1781   ∈ wcel 2111

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2729

This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1541  df-ex 1782  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830

This theorem is referenced by: (None)